This non-convexity can be detrimental when using these kernel loss functions to fit models by distance minimization, as in Figure REF .
We consider two discrete distributions \(_x = \tfrac{1}{N}\sum _{i=1}^N _{x_i}\)  and \(_y = \tfrac{1}{N}\sum _{i=1}^N _{y_i}\) .
We optimize \(_x\) , which is directly parameterized by the positions \(x=(x_1,\ldots ,x_N)\) , by doing a gradient descent of the energy \(x \mapsto (x) \triangleq {- }_{^*}^2\) .
This type of optimization is motivated by applications as diverse as the training of neural networks with a single hidden layer [1]}, [2]} (where the \(x_i\)  represent the hidden neurons), domain adaptation [3]} and optical flow estimation [4]}, [5]}.
We display in Figure REF  the initialization and the output of the gradient flow after performing gradient descent on \((x)\) .
One sees that for the Gaussian kernel (with small \(\sigma \) ), the output \(\)  does not match \(\) , with even particles moving away from the target \(\) .
For this task the Gaussian kernel is thus not efficient.
In contrast, the energy distance kernel is able to retrieve the support of \(\) , but the convergence is slow and particles are lagging behind.
This lag is problematic in training as it requires more gradient descent iterations (and thus it takes more time) to obtain the model's convergence. Our observations are only empirical and this kernel is not known to have better (Wasserstein) geodesic convexity than Gaussian kernels.
Leveraging OT, we show in Section REF  it is possible to avoid this issue (see Figure REF ), thus making it a powerful tool for model fitting tasks.
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